# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

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### Is a birational morphism between normal projective varieties residually separable?

My goal is to use a "normal Bertini" theorem (see https://link.springer.com/article/10.1007%2Fs000130050213)
More specifically, let $k$ be a field (you may assume that k is infinite but it should be ...

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### Can someone help in how to approach reading Mordell-Weil Theorem for abelian varieties?

I was thinking to start reading the proof of Mordell-Weil Theorem for abelian varieties over number fields, after getting done with the proof in the case of elliptic curves over number field and I ...

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### Bounded Torsion, without Mazur’s Theorem

Mazur’s torsion theorem famously tells us exactly which finite groups can occur as the torsion subgroup of $E(\mathbb{Q})$ for an elliptic curve $E$ defined over $\mathbb{Q}$. In particular, it ...

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### Can someone help in understanding this isomorphism helpful in proving finiteness of n-Selmer group?

This is from J.S.Milne's Elliptic Curves book.
We have $H^1(\mathbb{Q}, E_2) \cong (\mathbb{Q}^× / \mathbb{Q}^{×2})^2$ because $Gal(\mathbb{Q}^{al} / \mathbb{Q})$ acts trivially on $E_2(\mathbb{Q}^{...

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### Points on an Elliptic Curve, how to interpret $(x(2P):z(2P))$? [on hold]

This is from J.S. Milne's book 'Elliptic Curves'-
With $E(\mathbb{Q}) : Y^2Z = X^3 +aX Z^2+bZ^3$ and given a point on $P =(x,y)$ on it's Weierstrass equation dehomogenized $E: y^2= x^3+ax+b$ where $a,...

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### Having difficulty in understanding a result that'll help in proving the finiteness of Selmer group

I'm reading and trying to understand the proof of the finiteness of n-Selmer group from J.S.Milne's Elliptic Curves book but having difficulty in understanding it. Here's a screenshot from the book-
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272 views

### Fontaine-Fargues curve and period rings and untilt

When I read the paper "THE FARGUES–FONTAINE CURVE AND DIAMONDS" of Matthew Morrow, I have a question on page 11.
Question: The arthur said that the de Rham and crystalline period rings implicitly ...

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239 views

### How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?

Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...

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528 views

### Classify 2-dim p-adic galois representations

Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...

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### Complex isomorphism class of abelian varieties and $L$-functions

In his famous Mordell paper, Faltings proved that two abelian varietes $A_1, A_2$ defined over a number field $K$ are isogenous if and only if the local $L$-factors of $A_1, A_2$ are equal at every ...

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### Can someone help clear up some confusion regarding the proof of Mordell-Weil theorem?

I'm reading the proof(s) of Mordell-Weil theorem using various texts. This post is to make sure what I'm reading and understanding is correct..
Rational points on Elliptic Curves by Silverman Tate ...

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107 views

### Dimension of Prym variety of cover

I am reading the article by Lawrence and Venkatesh on diophantine problems and $p-$adic period mappings. At page $35$ they say that the dimension of the Prym variety of an (unramified) cover of curves ...

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68 views

### Tate modules of Jacobian varieties isomorphic over $\overline{\mathbb{Q}}$ but not over a number field $K$

Let $C_1, C_2$ be two curves defined over a number field $K$. Suppose that $C_1, C_2$ are isomorphic over $\overline{\mathbb{Q}}$ but not over $K$ and that $C_1(K), C_2(K) \ne \emptyset$. Then the ...

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394 views

### Is every positive integer the rank of an elliptic curve over some number field?

For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$?
It's easy to show that the set of such $n$ is unbounded. But can one show that ...

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106 views

### Which Shimura varieties admit or don't admit $p$-adic uniformization by Drinfeld spaces?

$p$-adic uniformization is a powerful tool for studying Shimura curves and Shimura varieties. For instance, we know cohomology groups of Drinfeld spaces, so we know some information about the Shimura ...